1. Field of the Invention
Embodiments of the invention generally relate to electronics, and in particular, to adaptive equalization of communications signals.
2. Description of the Related Art
Physical impairments can limit the effective transmission of data signals over communications channels. For example, channels can be frequency selective and can attenuate and phase shift the various frequency components of an input signal in a non-uniform manner, resulting in channel distortion. The corresponding impulse response of the channel can span several symbol intervals, which results in time-smearing and intersymbol interference (ISI). The ISI resulting from channel distortion, if left uncompensated, can cause high error rates.
One approach to handling ISI is to compensate for or reduce the ISI in the received signal with an equalizer. Various equalization techniques exist. For channels with relatively mild ISI impairments, a linear equalizer (LE) can be used. The linear equalizer is a sub-optimal equalizer structure implemented with a relatively simple finite impulse response (FIR) filter. It is popular because it has relatively low computational complexity compared to other equalizers, such as an equalizer based on optimal maximum likelihood sequence estimation (MLSE). The filter coefficients of a linear equalizer can be fixed and based on a known channel impulse response, or can be adaptively adjusted in response to channel characteristics, which can vary and can change over time. These linear equalizers are typically implemented digitally.
Many adaptive algorithms exist for finding relatively good or optimal equalizer tap coefficients depending on criteria, a-priori knowledge of channel characteristics, and the like. Adaptive algorithms include statistical methods for feature and model adaptation, which can be complex to implement. However, when the communication environment can be approximated with a Gaussian channel, the optimal equalizer taps can be determined by a relatively simple minimum mean-squared-error (MMSE) criterion. A linear MMSE receiver should minimize the error variance encountered at the slicer and consequently bit-error-rate (BER).
Minimum mean-squared-error (MMSE) is usually implemented with least-mean-square (LMS) algorithm, which is computationally efficient. Different variants of LMS such as block-LMS, leaky-LMS, sign (or clipped) LMS algorithms, normalized LMS, and the like, can alternatively be used. In this disclosure, reference to LMS adaptation includes all these variants.
Clipping is an example of a non-linear distortion that can severely impact linear equalizer performance. For example, an LMS algorithm is typically sensitive to a harsh non-linear distortion of the received signal. Unless clipping occurs rarely, such as less than 1% of the time, clipping will significantly reduce the performance of an LMS algorithm. Conventional techniques exist to avoid or ameliorate the effects of clipping.
For example, interpolation can be used. See, for example, U.S. Pat. No. 6,606,047 to Borjesson, et al. When dealing with the problem of equalizing clipped signals in OFDM systems that achieve reduced peak-to-average ratio by clipping and filtering the transmit signal, the effect of clipping noise at the receiver can be recreated to remove it from the incoming signal as presented in Bittner, S. et al. in Iterative Correction of Clipped and Filtered Spatially Multiplexed OFDM Signals, Proceedings of the 67th IEEE Vehicular Technology Conference, Spring 2008, pp. 953-957, May 2008.
Other systems attempt to avoid equalizer operation in a non-linear region. See U.S. Pat. No. 7,336,729 to Agazzi and U.S. Pat. No. 7,346,119 to Gorecki, et al.
These equalizers are commonly used in serializer-deserializer (SERDES) applications. One application of SERDES is to transfer data over a backplane channel at a relatively high data rate. FIG. 1 illustrates an example of a channel impulse response (CIR) of a typical backplane used for 6 Gbit/s operation.
Under the conditions of no correlations among the transmitted data, a Gaussian noise environment, and no non-linear distortions, the distortion from a channel having characteristics illustrated in FIG. 1 should be readily equalized with a linear minimum mean-squared-error (MMSE) equalizer using LMS adaptation. The ISI distortion of such a channel is not expected to be severe, and a relatively simple 2-tap adaptive equalizer as illustrated in FIG. 2 can be used.
The 2-tap adaptive equalizer illustrated in FIG. 2 is termed a post-cursor equalizer because the equalizer mostly affects the dominant post-cursors of the impulse response. The illustrated 2-tap adaptive equalizer includes a multiplier 202 for a first tap coefficient c0(k), a multiplier 204 for a second tap coefficient c1(k), a delay element 206, a summing circuit 208, a slicer 210, a differencing circuit 212, and an adaptation engine 214. In a high-speed SERDES, the taps of the equalizer are usually normalized so that the coefficient c0(k) for the first tap is equal to 1. In that instance, the 2-tap post-cursor equalizer illustrated in FIG. 2 is usually referred to as a single tap equalizer.
Assuming a Gaussian channel and an uncorrelated input sequence, the adaptation of the equalizer of FIG. 2 can be achieved using a least mean squares (LMS) algorithm. Applicable LMS adaptation equations are expressed in Equations 1(a) and 1(b).c0(k+1)=c0(k)+μ·e(k)·x(k)  (Eq. 1A)c1(k+1)=c1(k)+μ·e(k)·x(k−1)  (Eq. 1B)
In Equations 1A and 1B, the symbol μ represents the adaptation step, and k is the time index. The LMS adaptation of equalizer taps [c0, c1] is driven by the input samples [x(k), x(k−1)] and the error signal e(k). The input samples x(k) are soft, such as, quantized to 3 bits or more by an analog-to-digital converter, but can be clipped or compressed as will be explained later. Soft information carried by those signals and a low bit error rate (BER) at the slicer output (hard) permit proper convergence of the algorithm.
For the adaptive equalizer illustrated in FIG. 2, the error signal e(k) and soft equalizer output signal y(k) can be as expressed by Equations 2A and 2B.e(k)=d(k)−y(k)  (Eq. 2A)y(k)=c0(k)·x(k)+c1(k)·x(k−1)  (Eq. 2B)
Under ideal and near ideal conditions, an adaptive LMS algorithm will typically converge to a solution for the linear equalizer with optimal MMSE tap values (filter coefficients). For example, with the example channel illustrated in FIG. 1, the optimal equalizer tap coefficient values are [2.12, −0.80], or alternatively in the normalized single-tap form are [1, −0.38].
However, when the received signal x(k) is severely distorted by harsh compression or clipping, the soft information relied upon for correct convergence will typically not be available. Under clipped conditions, the normally “soft” received signal x(k) can be modeled by x_clip(k), which can have a “hard” characteristic, as illustrated in Equation 3.k×clip(k)=sign[x(k)]·nl_level  (Eq 3)
The variable nl_level of Equation 3 depends on the communications channel input/output transfer. When the non-linearity is described by a limiter, the variable nl_level is equal to the clip level. Depending on the analog-to-digital converter (ADC) clipping level and the channel characteristics, various clipping rates, that is, the rate at which samples are clipped, can be encountered. In addition, a 2-tap equalizer can have 4 possible combinations of clipped/not-clipped symbol samples for [x(k) x(k−1)] that drive the adaptation algorithm. These combinations are: both samples clipped, both samples not clipped, and only one or the other sample clipped.
When the equalizer adaptation is driven by clipped samples, the resulting tap values generated by adaptation can vary significantly from the values for an optimal MMSE receiver. For example, with the example channel of FIG. 1 and with the clipping level set so that the clipping rate is 0.5 or 50%, the LMS algorithm can deviate from an optimal linear MMSE solution and converge to equalizer tap coefficient values of [2.50, −0.63], or for the normalized single tap form, [1, −0.25]. In this example, the loss of soft information results in a considerable under-equalization of the samples and worse performance for ISI.